6,643 research outputs found
Analytic mappings between noncommutative pencil balls
In this paper, we analyze problems involving matrix variables for which we
use a noncommutative algebra setting. To be more specific, we use a class of
functions (called NC analytic functions) defined by power series in
noncommuting variables and evaluate these functions on sets of matrices of all
dimensions; we call such situations dimension-free.
In an earlier paper we characterized NC analytic maps that send
dimension-free matrix balls to dimension-free matrix balls and carry the
boundary to the boundary; such maps we call "NC ball maps". In this paper we
turn to a more general dimension-free ball B_L, called a "pencil ball",
associated with a homogeneous linear pencil L(x):= A_1 x_1 + ... + A_m x_m,
where A_j are complex matrices. For an m-tuple X of square matrices of the same
size, define L(X):=\sum A_j \otimes X_j and let B_L denote the set of all such
tuples X satisfying ||L(X)||<1.
We study the generalization of NC ball maps to these pencil balls B_L, and
call them "pencil ball maps". We show that every B_L has a minimal dimensional
(in a certain sense) defining pencil L'. Up to normalization, a pencil ball map
is the direct sum of L' with an NC analytic map of the pencil ball into the
ball. That is, pencil ball maps are simple, in contrast to the classical result
of D'Angelo on such analytic maps in C^m. To prove our main theorem, this paper
uses the results of our previous paper mentioned above plus entirely different
techniques, namely, those of completely contractive maps.Comment: 30 pages, final version. To appear in the Journal of Mathematical
Analysis and Application
Modelling the spectral energy distribution of galaxies from the ultraviolet to submillimeter
We present results from a new modelling technique which can account for the
observed optical/NIR - FIR/submm spectral energy distributions (SEDs) of normal
star-forming galaxies in terms of a minimum number of essential parameters
specifying the star-formation history and geometrical distribution of stars and
dust. The model utilises resolved optical/NIR images to constrain the old
stellar population and associated dust, and geometry-sensitive colour
information in the FIR/submm to constrain the spatial distributions of young
stars and associated dust. It is successfully applied to the edge-on spirals
NGC891 and NGC5907. In both cases the young stellar population powers the bulk
of the FIR/sub-mm emission. The model also accounts for the observed surface
brightness distribution and large-scale radial brightness profiles in NGC891 as
determined using the Infrared Space Observatory (ISO) at 170 and 200 mcrions
and at 850 micron using SCUBA.Comment: 20 pages (Latex), Highlight talk at the Joint European and National
Astronomical Meeting JENAM 2001, to be published in Reviews in Modern
Astronomy 15: Five Days of Creation: Astronomy with Large Telescopes from
Ground and Base. Germany : Astronomische Gesellschaft, 200
Noncommutative ball maps
In this paper, we analyze problems involving matrix variables for which we
use a noncommutative algebra setting. To be more specific, we use a class of
functions (called NC analytic functions) defined by power series in
noncommuting variables and evaluate these functions on sets of matrices of all
dimensions; we call such situations dimension-free. These types of functions
have recently been used in the study of dimension-free linear system
engineering problems.
In this paper we characterize NC analytic maps that send dimension-free
matrix balls to dimension-free matrix balls and carry the boundary to the
boundary; such maps we call "NC ball maps". We find that up to normalization,
an NC ball map is the direct sum of the identity map with an NC analytic map of
the ball into the ball. That is, "NC ball maps" are very simple, in contrast to
the classical result of D'Angelo on such analytic maps over C. Another
mathematically natural class of maps carries a variant of the noncommutative
distinguished boundary to the boundary, but on these our results are limited.
We shall be interested in several types of noncommutative balls, conventional
ones, but also balls defined by constraints called Linear Matrix Inequalities
(LMI). What we do here is a small piece of the bigger puzzle of understanding
how LMIs behave with respect to noncommutative change of variables.Comment: 46 page
Work extraction and thermodynamics for individual quantum systems
Thermodynamics is traditionally concerned with systems comprised of a large
number of particles. Here we present a framework for extending thermodynamics
to individual quantum systems, including explicitly a thermal bath and
work-storage device (essentially a `weight' that can be raised or lowered). We
prove that the second law of thermodynamics holds in our framework, and give a
simple protocol to extract the optimal amount of work from the system, equal to
its change in free energy. Our results apply to any quantum system in an
arbitrary initial state, in particular including non-equilibrium situations.
The optimal protocol is essentially reversible, similar to classical Carnot
cycles, and indeed, we show that it can be used it to construct a quantum
Carnot engine.Comment: 11 pages, no figures. v2: published version. arXiv admin note:
substantial text overlap with arXiv:1302.281
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